I want to know which of the following will under go dehydration most rapidly and why.

1) 3-hydroxy-2-hexanone
2) 4-hydroxy-2-hexanone
3) 5-hydroxy-2-hexanone
5) 3-hexanol

well, I can't give you a precise answer, but I can help you narrow it down.
(I assume you mean "will undergo most readily" as in, "this molecule is best suited to undergo dehydration"-- if I'm mistaken, let me know)

your 3-hydroxy-2-hexanone if dehydrated would create a double bond between either carbons 2 and 3, or between 3 and 4. However, since the second carbon already has all its electrons already bonded, this would be impossible. The bond between carbons 3 and 4 would be acceptable, still, but you limit the options available for this reaction by one.

the 4-hydroxy-2-hexanone is still in the running because its hydroxy-group is still secondary... dehydration will yield either an alkene bond between 3 and 4, or between 4 and 5. Either is acceptable.

the 5-hydroxy-2-hexanone I would say could be eliminated from the running just as #1 can be, becuase dehydration would yield an alkene bond between 5 and 6 as a potential product, but this is not the most highly substituted product. Remember the Saitzev/Zaitsev Rule.
This eliminates one of the possiblities for a dehydration product, so I would suggest one of your other options.

your 3-hexanol is still, I'd say, a pretty good bet. You'd still have the chance for two potential products with alkene bonds after dehydration.

If something about the carbonyl group is causing steric hinderance in your 4-hydroxy-2-hexanone, I'd say your best bet is probably the 3-hexanol, but I don't know if the steric hinderance would really play a role there as a determining factor...

hope that helped a bit to get you thinking.